Lemma 24.6.3Let u be a Lipschitz continuous function which vanishes outside somecompact set. Then there exists a unique u_{,i}∈ L^{∞}
(ℝn )
such that
lim u-(⋅+-h)-− u-(⋅) = u,i weak ∗ in L∞ (ℝn).
h→0 h
Proof:By the Lipschitz condition, the above difference quotient is bounded in L^{∞} by K
the Lipschitz constant of u. It follows from the Banach Aloglu theorem and Corollary
15.5.6 on Page 1329 that there exists a subsequence h_{k}→ 0 and g ∈ L^{∞}
(ℝn)
such
that
u(⋅+ hk)− u(⋅) ∞ n
-----hk-------→ g weak ∗ in L (ℝ )
Letting ϕ ∈ C_{c}^{∞}
(ℝn )
, it follows
∫ ∫ u(⋅+ h )− u(⋅) ∫
gϕdx = lim ------k-------ϕdx = − u ϕ,idx
k→ ∞ hk
This also shows that g must vanish outside some compact set because the integral on the
right shows that if sptϕ does not intersect sptu, then ∫gϕdx = 0. Thus g ∈ L^{2}
(ℝn)
. If g_{1}
is a weak ∗ limit of another subsequence h_{j}→ 0, the same result follows. Thus for any
ϕ ∈ C_{c}^{∞}
(ℝn)
∫
(g − g1)ϕdx = 0
and since C_{c}^{∞}
(ℝn)
is dense in L^{2}
(ℝn)
, this requires g = g_{1} in L^{2} and so they are equal
a.e. Since every sequence of h → 0 has a subsequence which when applied to the
difference quotient, always converges to the same thing, it follows the claimed limit exists.
This is called u_{,i}. This proves the lemma.
Lemma 24.6.4Let u be a Lipschitz continuous function which vanishes outside acompact set and let u_{,i}be described above. For ϕ_{ε}a mollifier and u_{ε}≡ u ∗ ϕ_{ε},
uε,i = u,i ∗ϕε
where the symbol u_{ε,i}means the usual partial derivative with respect to the i^{th}variable.Also for any p > n,
uε,i → u,i in Lp (ℝn ).
Proof: This follows from a computation and Lemma 24.6.3.
∫ u(x− y + he )− u(x− y)
uε,i(x) ≡ lim -----------i----------ϕ ε(y )dy
h→0 h
. But also you could take w = x in 24.6.18 which
yields h
(x)
≤ h
(x )
. Therefore h
(x)
= h
(x)
if x ∈ Ω.
Now suppose x,y∈ ℝ^{n} and consider
||- -- ||
h (x )− h(y)
. Without loss of generality
assume h
(x)
≥h
(y)
.(If not, repeat the following argument with x and y interchanged.)
Pick w ∈ Ω such that
--
h (w )+ K |y− w |− ε < h (y) .
Then
|-- -- | -- --
|h(x)− h (y )| = h (x)− h(y) ≤ h (w)+ K |x − w |−
[h(w )+ K |y − w |− ε] ≤ K |x − y |+ ε.
Since ε is arbitrary,
|-- -- |
|h(x)− h(y)| ≤ K |x− y |
and this proves the theorem.
With this theorem, here is the main result called Rademacher’s theorem.
Theorem 24.6.7Let h : Ω → ℝ^{m}be Lipschitz on Ω where Ω is some nonemptymeasurable set in ℝ^{n}. Then Dh
(x)
exists for a.e. x ∈ Ω. If Ω = ℝ^{n}, then for eache_{i},
lim h(⋅+-hei)−-h(⋅)= h weak ∗ in L∞ (ℝn)
h→0 h ,i
and whenever ϕ_{ε}is a mollifier,
p n m
(h∗ ϕε),i → h,i in L (ℝ ;ℝ ).
Proof:The last two claims follow from the above argument applied to the
components of h. By Theorem 24.6.6 the function can be extended to a Lipschitz
function defined on all of ℝ^{n}, still denoted as h. Let Ω_{r}≡ Ω ∩ B
(0,r)
. Now let
ψ ∈ C_{c}^{∞}
(B (0,2r))
such that ψ = 1 on B
( 3 )
0,2r
. Then ψh is Lipschitz on ℝ^{n}
and vanishes off a bounded set. It follows from Lemma 24.6.5 applied to the
components of h that this function has a derivative off a set of measure zero
N_{r}. If x ∈ Ω_{r}∖ N_{r} it follows since ψ = 1 near x that Dh
(x)
exists. Letting
N = ∪_{r=1}^{∞}N_{r}, it follows that if x ∈ Ω ∖ N, then Dh
(x)
exists. This proves the
theorem.
For u Lipschitz as described above, the limit of the difference quotient u_{,i} is called the
weak partial derivative of u. For p > n and an assertion that the difference quotients are
bounded in L^{p} everything done above would work out the same way and one can
therefore generalize parts of the above theorem. The extension is problematic
but one can give the following results with essentially the same proof as the
above.
Lemma 24.6.8Let u ∈ L^{p}
(ℝn)
. There exists u_{,i}∈ L^{p}
(ℝn)
such that
u(⋅+-hei)-− u-(⋅) p n
lhim→0 h = u,i weakly in L (ℝ )
if and only if the difference quotients
u(⋅+hei)−u(⋅)
h
are bounded in L^{p}
(ℝn )
for all nonzeroh.
Proof: If the weak limit exists, then the difference quotients must be bounded. This
follows from the uniform boundedness theorem, Theorem 15.1.8. Here is why. Denote the
difference quotient by D_{h} to save space. Weak convergence requires ∫D_{h}f →∫u_{.i}f for
all f ∈ L^{p′
}. Could there exist h_{k} such that
||Dhk||
_{Lp}→∞? Not unless a subsequence
satisfies h_{k}→ 0 because if this sequence is bounded away from 0, the formula for D_{h} will
yield the difference quotients are bounded. However, if h_{k}→ 0, then for each
f ∈ L^{p′
},
∫
sup D f < ∞
k hk
because in fact, lim_{k→∞}∫D_{hk}f exists so it must be bounded. Now D_{hk} can be
considered in
(Lp′)
^{′} and this shows it is pointwise bounded on L^{p′
}. Therefore, D_{hk} is
bounded in
( ′)
Lp
^{′} but the norm on this is the same as the norm in L^{p}. Thus D_{hk} is
bounded after all.
Conversely, if the difference quotients are bounded, the same argument used earlier,
involving convergence of a subsequence, this time coming from the Eberlein Smulian
theorem, Theorem 15.5.12 and showing that every subsequence converges to the same
thing, shows the difference quotients converge weakly in L^{p}
(ℝn)
to something we can call
u_{,i}. This proves the lemma.
Definition 24.6.9A function f ∈ L^{p}
(ℝn)
is said to have weak partial derivativesin L^{p}
(ℝn )
if the difference quotients
u(⋅+hei)−u(⋅)
h
for each i = 1,2,
⋅⋅⋅
,n arebounded for h≠0. If f ∈ L^{p}
(ℝn; ℝm)
, it has weak partial derivatives in L^{p}
(ℝn; ℝm)
if each component function has weak partial derivatives in L^{p}
(ℝn )
.
This following theorem may also be referred to as Rademacher’s theorem.
Theorem 24.6.10Let h be in L^{p}
(ℝn;ℝm )
,p > n, and suppose it has weak derivativesh_{,i}∈ L^{p}
(ℝn;ℝm )
for i = 1,
⋅⋅⋅
,n. Then Dh
(x)
exists a.e. and h is almosteverywhereequal to a continuous function. Also if ϕ_{ε}is a mollifier,
_{,i}→ h_{,i} follows as before from a use of Minkowski’s inequality. Letting
u be one of the component functions of h, Morrey’s inequality holds for u_{ε}≡ u ∗ ϕ_{ε}.
Thus
(∫ )1∕p( )
|uε(x)− uε(y)| ≤ C |∇uε (z)|pdz |x− y|1−n∕p
B (x,2|x−y|)
Now there exists a subsequence such that u_{ε}→ u pointwise a.e. and also each
u_{ε,i}→ u_{,i} pointwise a.e. as well as in L^{p}. Therefore, for x,y not in a set of measure
zero,
( )1 ∕p
∫ p ( 1− n∕p)
|u(x)− u (y)| ≤ C B(x,2|x−y|)|∇u (z)| dz |x − y|
which shows the claim about u being equal to a continuous function off a set of measure
zero. Thus h is also continuous off a set of measure zero.